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Computational Approach to Riemann Surfaces

  • Book
  • © 2011

Overview

Editors:
  1. Alexander I. Bobenko

    1. Institute of Mathematics, Technical University of Berlin, Berlin, Germany

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  2. Christian Klein

    1. Institut de Mathématiques, Université de Bourgogne, Dijon Cedex, France

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    You can also search for this editor in PubMed   Google Scholar

  • Self-contained introduction to the theory of Riemann surfaces
  • Detailed explanation of existing codes with examples
  • Visualization of solutions to integrable partial differential equations and surfaces
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2013)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    1. Front Matter

      Pages 1-1
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    2. Introduction to Compact Riemann Surfaces

      • Alexander I. Bobenko
      Pages 3-64

  3. Algebraic Curves

    1. Front Matter

      Pages 65-65
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    2. Computing with Plane Algebraic Curves and Riemann Surfaces: The Algorithms of the Maple Package “Algcurves”

      • Bernard Deconinck, Matthew S. Patterson
      Pages 67-123

    3. Algebraic Curves and Riemann Surfaces in Matlab

      • Jörg Frauendiener, Christian Klein
      Pages 125-162

  4. Schottky Uniformization

    1. Front Matter

      Pages 163-163
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    2. Computing Poincaré Theta Series for Schottky Groups

      • Markus Schmies
      Pages 165-182

    3. Uniformizing Real Hyperelliptic M-Curves Using the Schottky–Klein Prime Function

      • Darren Crowdy, Jonathan S. Marshall
      Pages 183-193

    4. Numerical Schottky Uniformizations: Myrberg’s Opening Process

      • Rubén A. Hidalgo, Mika Seppälä
      Pages 195-209

  5. Discrete Surfaces

    1. Front Matter

      Pages 211-211
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    2. Period Matrices of Polyhedral Surfaces

      • Alexander I. Bobenko, Christian Mercat, Markus Schmies
      Pages 213-226

    3. On the Spectral Theory of the Laplacian on Compact Polyhedral Surfaces of Arbitrary Genus

      • Alexey Kokotov
      Pages 227-253

  6. Back Matter

    Pages 255-257
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Keywords

About this book

This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development. The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.e., calculus and linear algebra, is required. In particular, no knowledge of the theory of Riemann surfaces is expected; the necessary background in this theory is contained in the Introduction chapter. At the same time, this book is also intended for specialists in geometry and mathematical physics applying the theory of Riemann surfaces in their research. It is the first book on numerics of Riemann surfaces that reflects the progress made in this field during the last decade, and it contains original results. There are a growing number of applications that involve the evaluation of concrete characteristics of models analytically described in terms of Riemann surfaces. Many problem settings and computations in this volume are motivated by such concrete applications in geometry and mathematical physics.

Editors and Affiliations

  • Institute of Mathematics, Technical University of Berlin, Berlin, Germany

    Alexander I. Bobenko

  • Institut de Mathématiques, Université de Bourgogne, Dijon Cedex, France

    Christian Klein

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